\newproblem{lay:4_5_25}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.5.25}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $S$ be a subset of an $n$-dimensional vector space $V$, and suppose $S$ contains fewer than $n$ vectors. Explain why $S$ cannot span $V$.
}{
  % Solution
	Note that $n\geq 1$ because $S$ cannot have fewer than 0 vectors. If $S$ spans $V$, then there exists a subset $S'\subseteq S$ that is a basis of $V$. $S'$ must have fewer than $n$
	vectors (because $S$ has fewer than $n$ vectors), but by Theorem 9.2 of Chapter 5, this is impossible because all bases of $V$ have $n$ vectors.
}
\useproblem{lay:4_5_25}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
